I've created a procedurally animated Rorschach test but I don't know how to stop it from being cut off on the edges?
The key to prevent the texture from cutting off at the edges is to reduce its density towards the edges. The simplest way of achieving this is to use the Distance (Vector Math node) from the centre to adjust the output from the Musgrave texture. By using some maths functions we can change how quickly and how sharply the adjustment falls off as the point moves away from the centre.
I've replicated your nodes and added the ones highlighted to apply a factor based on the distance :

So the Distance from the origin is first passed through a Power node. The higher the power, the faster the falloff (so the sharper the edge). The multiply node allows the 'strength' to be adjusted and this is combined with the output of the Musgrave.
This produces the following result :